Problem 29 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{\left(m^{5}\right)^{4} m}{m^{10}} $$

Step-by-Step Solution

Answer

The simplified expression is \(m^{11}\).

Step 1: Apply the Power of a Power Rule

We start by applying the Power of a Power Rule to \((m^5)^4\). According to the rule, \((a^m)^n = a^{m imes n}\). Therefore, we can simplify \((m^5)^4\) to \(m^{5 imes 4} = m^{20}\). Now, our expression becomes \(\frac{m^{20} \cdot m}{m^{10}}\).

Step 2: Use the Product of Powers Rule

Next, we use the Product of Powers Rule, which states that \(a^m \cdot a^n = a^{m+n}\). In our expression, we have \(m^{20} \cdot m\). So, this simplifies to \(m^{20 + 1} = m^{21}\). Now, the expression is \(\frac{m^{21}}{m^{10}}\).

Step 3: Apply the Quotient of Powers Rule

Now, we use the Quotient of Powers Rule, which tells us that \(\frac{a^m}{a^n} = a^{m-n}\), provided \(a eq 0\). Applying this to \(\frac{m^{21}}{m^{10}}\), we get \(m^{21-10} = m^{11}\).

Step 4: Confirm Exponent is Positive

Finally, we check to ensure the exponent in \(m^{11}\) is positive. Since 11 is already positive, we have successfully simplified the expression.

Key Concepts

Power of a Power Rule

The Power of a Power Rule is an essential concept in algebra, especially when dealing with exponents. When you have a power raised to another power, such as \((a^m)^n\), this rule helps you simplify the expression.

The formula is: \( (a^m)^n = a^{m \times n} \).
This means you multiply the exponents.

• In the expression \((m^5)^4\), multiply \(5\) by \(4\) to get \(m^{20}\).

Once you apply this rule, you've reduced the complexity of the original expression considerably. Understanding this step is crucial, as it lays the foundation for utilizing other exponent rules effectively.

Product of Powers Rule

The Product of Powers Rule is useful when multiplying two expressions that have the same base. This rule states that \(a^m \cdot a^n = a^{m+n}\).
When you multiply powers with the same base, you simply add the exponents.

• For example, in the expression \(m^{20} \cdot m\), you add the exponents \(20\) and \(1\) to get \(m^{21}\).

This rule is helpful for simplifying expressions and reducing long multiplication processes to simpler addition. It prepares you for further simplification using division, where the Quotient of Powers Rule comes into play.

Quotient of Powers Rule

To simplify algebraic expressions involving division, the Quotient of Powers Rule is applied. This rule states \(\frac{a^m}{a^n} = a^{m-n}\) provided \(a eq 0\).
When dividing powers with the same base, subtract the exponents.

• In the expression \(\frac{m^{21}}{m^{10}}\), subtract \(10\) from \(21\) to simplify it to \(m^{11}\).

Remember, this rule allows you to break down complex fractions into simple expressions. Ensuring the base is not zero is crucial, as division by zero is undefined.

Positive Exponents

Having positive exponents in your final solution is important for clarity and standard form. Positive exponents indicate straightforward multiplication, while negative exponents suggest inverse relationships (like division).

In the problem, the expression \(m^{11}\) has a positive exponent, which is ideal.

• Always aim to express final results with positive exponents. It simplifies interpreting and using the expression in further calculations or real-world applications.
• In scenarios where a negative exponent appears, convert it to positive by expressing the inverse.

Focusing on positive exponents nurtures a better understanding of powers and supports simplifying algebraic expressions effectively.